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Definition of Inverse function: Suppose that $X$ and $Y$ are sets, and $f \colon X \rightarrow Y$ is a function. Suppose that $g \colon Y \rightarrow X$ is another function. There are three notions of "inverse function", defined as follows:
- We say that $g$ is a left inverse of $f$ if $g \circ f = Id_X$ (the identity function from $X$ to itself).
- We say that $g$ is a right inverse of $f$ if $f \circ g = Id_Y$ (the identity function from $Y$ to itself).
- We say that $g$ is a two-sided inverse (or just inverse) of $f$ if $g$ is a left and right inverse.
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